In Lang's algebra I see the following definition:
"Let $G$ be a finite cyclic group of order $n$, generated by an element $\sigma$. Assume that $G$ operates on a an abelian group $A$ and let $f:A\to A$ be the endomorphism of $A$ given by $f(x)=\sigma x - x$ ..."
Why is $f$ an endomorphism? I thought $\sigma$ is just a permutation on $A$ and we don't have $\sigma(x+y)=\sigma x + \sigma y$.
Edit--The question Lang's Algebra: Herbrand quotient has Lang's exercise written out fully with some slight corrections, but also the assumption $\sigma(x+y)=\sigma x + \sigma y$, so there is probably some typo or ambiguity in the definitions. I just wanted to make sure the homomorphism property didn't somehow follow from the $G$ being a finite cylclic group and/or $A$ being abelian.